The following is the plot of my Ising Model simulation:

enter image description here

The upper one is for $T=3.3$, and the lower one is for $T=1.5$. Otherwise, in both cases, boltzmann=$1$, MC steps=$1000000$, and the lattice dimension are $40\times40$.

How can I identify phase transition in this plot?


1 Answer 1


For a having a phase transitions, small changes in parameters like temperature or pressure causes a large-scale, qualitative change in the state of a system.

To characterize the state of the system, you can use several thermodynamic variables like internal energy, magnetization, specific heat, etc.

The figure below show a phase transition:

sssss Caption: Ising phase transition as seen from classical Monte Carlo simulations with t′=0.15t and U=10t. Panels (a), (b), and (c) show the temperature dependence of the magnetization, susceptibility, and Binder cumulant, respectively. The insets are zoomed-in plots of the same data over a narrower temperature range. The temperature is plotted in units of J=4t2/U, with kB taken to be unity. Panels (d), (e), and (f) show the same data after scaling using the critical exponents of the 2D Ising model.

Reference: M. Karmakar and R. Ganesh. "Attractive Hubbard model as an SO(3) system of competing phases: Supersolid order and its thermal melting". Phys. Rev. Research 2, 023304, 2020. DOI: 10.1103/PhysRevResearch.2.023304.

Your plot is Energy vs Steps, but to detect phase transition, you need to plot Energy vs Temperature, for example. You can get an average energy for the last N steps for several temperatures and make a plot, then look for large variations in Energy.

  • $\begingroup$ What is $U$ here? The total energy? $\endgroup$
    – user366312
    Jan 29, 2023 at 17:21
  • $\begingroup$ What do u mean by $4t2/U$? $\endgroup$
    – user366312
    Jan 29, 2023 at 17:24
  • 1
    $\begingroup$ Please, read the publication. It is Open Access. The used Hamiltonian is shown in Eq. (1). $\endgroup$
    – Camps
    Jan 30, 2023 at 10:54

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